Find all real numbers \(k\) such that \(x^4+kx^3+x^2+4kx+16=0\) is true for exactly one real number \(x\). Note that there can be complex solutions as well. What are all possible values of \(k\)?

The possible values of k are 11/2 and -11/2.

Thanks! Can you show me how you arrived at that solution so that I will be able to solve similar problems on my own?